Problem: $f(x) = -4x^{2}-1$ What is the range of $f(x)$ ?
Consider the range of $-4x^{2}$ The range of $x^2$ is $\{\, y \mid y \ge 0 \,\}$ Multiplying by $-4$ flips the range to $\{\, y \mid y \le 0 \,\}$ To get $-4x^{2}-1$, we subtract $1$. So the range becomes: $\{\, y \mid y ≤ -1 \,\}$.